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Decidability results for ATL∗with imperfect information and

perfect recall

Raphaël Berthon

ENS Rennes

Rennes, France

raphael.berthon@ens-rennes.fr

Bastien Maubert

University of Naples Federico II

Naples, Italy

bastien.maubert@gmail.com

Aniello Murano

University of Naples Federico II

Naples, Italy

murano@na.infn.it

ABSTRACT

Alternating-time Temporal Logic (ATL∗) is a central logic

for multiagent systems. It has been extended in various

ways, notably with imperfect information (ATL∗

i). Since the

model-checking problem against ATL∗

ifor agents with per-

fect recall is undecidable, studies have mostly focused either

on agents without memory, or on alternative semantics to

retrieve decidability. In this work, we establish new, strong

decidability results for agents with perfect recall. We ﬁrst

prove a meta-theorem that allows the transfer of decidability

results for classes of multiplayer games with imperfect infor-

mation, such as games with hierarchical observation, to the

model-checking problem for ATL∗

i. We also establish that

model checking ATL∗with strategy context and imperfect

information for hierarchical instances is decidable.

1. INTRODUCTION

In formal system veriﬁcation, model checking is a well-

established method to automatically check the correctness

of a system [7, 30, 8]. It consists in modelling the system as

a mathematical structure, expressing its desired behaviour

as a formula from some suitable logic, and checking whether

the model satisﬁes the formula. In the nineties, interest has

arisen in the veriﬁcation of multiagent systems (MAS), in

which various entities (the agents) interact and can form

coalitions to attain certain objectives. This led to the devel-

opment of logics that allow reasoning about strategic abili-

ties in MAS [2, 26, 18, 34, 1, 6].

Alternating-time Temporal Logic (ATL∗), introduced by

Alur, Henzinger, and Kupferman [2], plays a central role in

this line of work. This logic, interpreted on concurrent game

structures, extends CTL∗with strategic modalities. These

modalities allow one to reason about the existence of strate-

gies for coalitions of agents to force the system’s behaviour to

satisfy certain temporal properties. ATL∗has been extended

in many ways, and among these extensions an important one

is ATL∗with strategy context [5, 24]. In ATL∗, strategies of

all agents are forgotten at each new strategic modality. In

ATL∗with strategy context (ATL∗

sc) instead they are stored

in a strategy context, and are forgotten only when replaced

by a new strategy or when the formula explicitly unbinds

the agent from her strategy. Thanks to this additional ex-

Appears in: Proceedings of the 16th International Confer-

ence on Autonomous Agents and Multiagent Systems (AA-

MAS 2017), S. Das, E. Durfee, K. Larson, M. Winikoﬀ

(eds.), May 8–12, 2017, S˜ao Paulo, Brazil.

Copyright c

2017, International Foundation for Autonomous Agents and

Multiagent Systems (www.ifaamas.org). All rights reserved.

pressive power, ATL∗

sc can express important game theoretic

concepts such as the existence of Nash Equilibria [24].

In many real-life scenarios, such as poker for instance,

agents do not always know precisely what is the current

state of the system. Instead, they have a partial view, or

observation, of the state. This fundamental feature of MAS

is called imperfect information, and it is known to quickly

bring about undecidability when involved in strategic prob-

lems, especially when agents have perfect recall of the past,

which is a usual and important assumption in games with

imperfect information and epistemic temporal logics [12].

For instance solving multiplayer games with imperfect in-

formation and perfect recall, i.e., deciding the existence of

a distributed winning strategy in such games, is already un-

decidable for reachability objective, as proven by Peterson,

Reif and Azhar [27]. Since such games are easily captured

by ATL∗with imperfect information (ATL∗

i), model checking

ATL∗

iwith perfect recall is also undecidable [2].

However it is known that restricting attention to cases

where some sort of hierarchy exists on the diﬀerent agents’

information yields decidability for several problems related

to the existence of strategies. Synthesis of distributed sys-

tems, which implicitly uses perfect recall and is undecid-

able in general [29], is decidable for hierarchical architec-

tures [22]. Actually, for branching-time speciﬁcations, dis-

tributed synthesis is decidable exactly on architectures free

from information forks, for which the problem can be re-

duced to the hierarchical case [13]. For richer speciﬁcations

from alternating-time logics, being free of information forks

is no longer suﬃcient, but distributed synthesis is decid-

able precisely on hierarchical architectures [31]. Similarly,

solving multiplayer games with imperfect information and

perfect recall, i.e., checking for the existence of winning dis-

tributed strategies, is decidable for ω-regular winning con-

ditions when there is a hierarchy among players, each one

observing more than those below [28, 22]. Recently, it has

been proven that this assumption can be relaxed while main-

taining decidability: the problem remains decidable if the hi-

erarchy can change along a play, or even if transient phases

without such a hierarchy are allowed [4].

Our contribution. In this work we establish several

decidability results for model checking ATL∗

iwith perfect

recall, with and without strategy context, all related to no-

tions of hierarchy. Our ﬁrst result is a theorem that allows

the transfer of decidability results for classes of multiplayer

games with imperfect information, such as those mentioned

above, to the model-checking problem for ATL∗

i. This theo-

rem essentially states that if solving multiplayer games with

imperfect information, perfect recall and omega-regular ob-

jectives is decidable on some class of concurrent game struc-

tures, then model checking ATL∗

iwith perfect recall is also

decidable on this class of models (a simple bottom-up algo-

rithm that evaluates innermost strategic modalities in every

state of the model suﬃces). As a direct consequence we eas-

ily obtain new decidability results for the model checking of

ATL∗

ion several classes of concurrent game structures.

Our second contribution considers ATL∗with imperfect

information and strategy context (ATL∗

sc,i). Because there

are in general inﬁnitely many possible strategy contexts, the

bottom-up approach used for ATL∗

icannot be used here.

Instead we build upon the proof presented in [24] that es-

tablishes the decidability of model checking ATL∗

sc by re-

duction to the model-checking problem for Quantiﬁed CTL∗

(QCTL∗). The latter extends CTL∗with second-order quan-

tiﬁcation on atomic propositions, and it has been well stud-

ied [33, 20, 21, 14, 23]. QCTL∗

i, an imperfect-information

extension of QCTL∗, has recently been introduced, and its

model-checking problem was proven decidable for the class

of hierarchical formulas [3]. In this paper we deﬁne a no-

tion of hierarchical instances for the ATL∗

sc,i model-checking

problem: informally, an ATL∗

sc,i formula ϕtogether with

a concurrent game structure Gis a hierarchical instance

if outermost strategic modalities in ϕconcern agents who

observe less in G. We adapt the proof from [24] and re-

duce the model-checking problem for ATL∗

sc,i on hierarchi-

cal instances to the model-checking problem for hierarchical

QCTL∗

iformulas. We obtain that model checking hierarchi-

cal instances of ATL∗

sc,i with perfect recall is decidable.

Related work. The model-checking problem for ATL∗

iis

known to be decidable when agents have no memory [32],

and the case of agents with bounded memory reduces to

that of no memory. Another way to retrieve decidability

is to assume that all agents in a coalition have the same

information, either because their observations of the system

are the same, or because they can communicate and share

their observations [15, 10, 16, 19]. This idea was also used

recently to establish a decidability result for ATL∗

sc,i [25]

when all agents have the same observation of the game.

The results we establish here thus strictly extend previ-

ously known results on the decidability of model checking

ATL∗

iand ATL∗

sc,i with perfect recall and standard seman-

tics, and they hold for vast, natural classes of instances, that

all rely on notions of hierarchy, which seems to be inherent to

all decidable cases of strategic problems for multiple entities

with imperfect information and perfect recall.

Outline. After setting some basic deﬁnitions in Section 2,

we present our meta-theorem on the model checking problem

for ATL∗

iin Section 3. In Section 4 we prove that when

restricted to hierarchical instances, model checking ATL∗

sc,i

is decidable, and we conclude in Section 5.

2. PRELEMINARIES

Let Σ be an alphabet. A ﬁnite (resp. inﬁnite )word over

Σ is an element of Σ∗(resp. Σω). The empty word is noted

, and Σ+= Σ∗\ {}. The length of a word is |w|:= 0 if w

is the empty word , if w=w0w1. . . wnis a ﬁnite nonempty

word then |w|:= n+ 1, and for an inﬁnite word wwe let

|w|:= ω. Given a word wand 0 ≤i, j ≤ |w| − 1, we let wi

be the letter at position iin wand w[i, j ] be the subword of

wthat starts at position iand ends at position j. For n∈N

we let [n] := {1,...,n}. Finally, for the rest of the paper, let

us ﬁx a countably inﬁnite set of atomic propositions AP and

let AP ⊂ AP be some ﬁnite subset of atomic propositions.

2.1 Kripke structures

AKripke structure over AP is a tuple S= (S, R, `) where

Sis a set of states,R⊆S×Sis a left-total1transition

relation and `:S→2AP is a label ling function.

Apointed Kripke structure is a pair (S, s) where s∈ S. A

path in a structure S= (S, R, `) is an inﬁnite word λover S

such that for all i∈N, (λi, λi+1)∈R. For s∈S, Paths(s)

is the set of all paths that start in s.

2.2 Inﬁnite trees

Let Xbe a ﬁnite set. An X-tree τis a nonempty set of

words τ⊆X+such that

•there exists r∈X, called the root of τ, such that each

u∈τstarts with r;

•if u·x∈τwith x∈Xand u6=, then u∈τ, and

•if u∈τthen there exists x∈Xsuch that u·x∈τ.

The elements of a tree τare called nodes. If u·x∈τ, we

say that u·xis a child of u. Similarly to Kripke structures,

apath is an inﬁnite sequence of nodes λ=u0u1. . . such that

for all i,ui+1 is a child of ui, and P aths(u) is the set of paths

that start in node u. An AP-labelled X-tree, or (AP, X)-tree

for short, is a pair t= (τ, `), where τis an X-tree called the

domain of tand `:τ→2AP is a labelling.

Definition 1 (Tree unfoldings). Let S= (S, R, `)

be a Kripke structure over AP, and let s∈S. The tree-

unfolding of Sfrom sis the (AP, S)-tree tS(s)=(τ, `0),

where τis the set of all ﬁnite paths that start in s, and for

every u∈τ,`0(u) = `(s), where sis the last letter of u.

3. ATL∗WITH IMPERFECT INFORMATION

In this section we recall the syntax and semantics of ATL∗

with imperfect information and synchronous perfect-recall

semantics, or ATL∗

ifor short, and establish a meta-theorem

on the decidability of its model-checking problem.

3.1 Deﬁnitions

We ﬁrst introduce the models of the logics we study. For

the rest of the paper, let us ﬁx a non-empty ﬁnite set of

agents Ag and a non-empty ﬁnite set of moves M.

Definition 2. Aconcurrent game structure with imper-

fect information (or CGSifor short) over AP is a tuple

G= (V, E , `, {∼a}a∈Ag)where Vis a non-empty ﬁnite set

of positions,E:V×MAg →Vis a transition function,

`:V→2AP is a labelling function and for each agent

a∈Ag,∼a⊆V×Vis an equivalence relation.

In a position v∈V, each agent achooses a move ma∈M,

and the game proceeds to position E(v, m), where m∈MAg

stands for the joint move (ma)a∈Ag (note that we assume

E(v, m) to be deﬁned for all vand m2). For each position

v∈V,`(v) is the ﬁnite set of atomic propositions that hold

in v, and for a∈Ag, equivalence relation ∼arepresents the

1i.e., for all s∈S, there exists s0such that (s, s0)∈R.

2This assumption, as well as the choice of a unique set of

moves for all agents, is made to ease presentation. All the

results presented here also hold when the set of available

moves depends on the agent and the position.

observation of agent a: for two positions v, v 0∈V,v∼av0

means that agent acannot tell the diﬀerence between vand

v0. We may write v∈ G for v∈V. A pointed CGSi(G, v) is

a CGSiGtogether with a position v∈ G.

In Section 3.2 we also use nondeterministic CGSi, which

are as in Deﬁnition 2 except that they have a transition re-

lation E⊆V×MAg ×Vinstead of a transition function. In

a position v, after every agent has chosen a move, forming

a joint move m∈MAg, a special agent called Nature (not

in Ag) chooses a next position v0such that (v, m, v0)∈E

(see [4] for detail). In the following, unless explicitly speci-

ﬁed, CGSialways refers to deterministic CGSi. The follow-

ing deﬁnitions also concern deterministic CGSi, but they

can be adapted to nondeterministic ones in an obvious way.

Aﬁnite (resp. inﬁnite)play is a ﬁnite (resp. inﬁnite)

word ρ=v0. . . vn(resp. π=v0v1. . .) such that for all i

with 0 ≤i < |ρ| − 1 (resp. i≥0), there exists a joint move

msuch that E(vi,m) = vi+1. A ﬁnite (resp. inﬁnite) play

ρ(resp. π)starts in a position vif ρ0=v(resp. π0=v).

We let Plays(G, v) be the set of plays, either ﬁnite or inﬁnite,

that start in v.

In this work we consider agents with synchronous perfect

recall, meaning that the observational equivalence relation

for each agent ais extended to ﬁnite plays the following way:

ρ∼aρ0if |ρ|=|ρ|0and ρi∼aρ0

ifor every i∈ {0,...,|ρ|−1}.

Astrategy for agent a is a function σ:V+→M such that

σ(ρ) = σ(ρ0) whenever ρ∼aρ0. The latter constraint cap-

tures the essence of imperfect information, which is that

agents can base their strategic choices only on the informa-

tion available to them, and removing this constraint yields

the semantics of classic ATL with perfect information.

Astrategy proﬁle for a coalition A⊆Ag is a mapping σA

that assigns a strategy to each agent a∈A; for a∈A, we

may write σainstead of σA(a). An inﬁnite play πfollows

a strategy proﬁle σAfor a coalition Aif for all i≥0, there

exists a joint move msuch that E(πi,m) = πi+1 and for

each a∈A,ma=σa(π[0, i]). For a strategy proﬁle σAand

a position v∈V, we deﬁne the outcome Out(v, σA) of σA

in vas the set of inﬁnite plays that start in vand follow σA.

The syntax of ATL∗

iis the same as that of ATL∗, and is

given by the following grammar:

ϕ::= p| ¬ϕ|ϕ∨ϕ| hAiϕ|Xϕ|ϕUϕ,

where p∈ AP and A⊆Ag.

Xand Uare the classic next and until operators, re-

spectively, while the strategic operator hAiquantiﬁes over

strategy proﬁles for coalition A.

The semantics of ATL∗

iis deﬁned with regards to a CGSi

G= (V, E , `, {∼a}a∈Ag), an inﬁnite play πand a position

i≥0 along this play, by induction on formulas:

G, π, i |=pif p∈`(πi)

G, π, i |=¬ϕif G, π, i 6|=ϕ

G, π, i |=ϕ∨ϕ0if G, π, i |=ϕor G, π, i |=ϕ0

G, π, i |=hAiϕif there exists a strategy proﬁle σAs.t.

for all π0∈Out(πi, σA), G, π0,0|=ϕ

G, π, i |=Xϕif G, π, i + 1 |=ϕ

G, π, i |=ϕUϕ0if there exists j≥is.t. G, π, j |=ϕ0and,

for all ks.t. i≤k < j,G, π, k |=ϕ.

An ATL∗

iformula ϕis closed if every temporal operator

(Xor U) in ϕis in the scope of a strategic operator hAi.

Since the semantics of a closed formula ϕdoes not depend on

the future, we may write G, v |=ϕ, meaning that G, π, 0|=ϕ

for any inﬁnite play πthat starts in v.

The model-checking problem for ATL∗

iconsists in deciding,

given a closed ATL∗

iformula ϕand a ﬁnite pointed CGSi

(G, v), whether G, v |=ϕ.

3.2 Model checking ATL∗

i

It is well known that the model-checking problem for ATL∗

i

is undecidable for agents with perfect recall [2], as it can eas-

ily express the existence of distributed winning strategies

for multiplayer reachability games with imperfect informa-

tion and perfect recall, which was proved undecidable by

Peterson, Reif and Azhar [27]. A direct proof of this unde-

cidability result for ATL∗

iis also presented in [11]. However,

there are classes of multiplayer games with imperfect infor-

mation that are decidable. For many years, the only known

decidable case was that of hierarchical games, in which there

is a total preorder among players, each player observing at

least as much as those below her in this preorder [28, 22].

Recently, this result has been extended by relaxing the as-

sumption of hierarchical observation. In particular, it has

been shown that the problem remains decidable if the hier-

archy can change along a play, or if transient phases without

such a hierarchy are allowed [4]. We establish that these re-

sults transfer to the model-checking problem for ATL∗

i.

We remind that a concurrent game with imperfect infor-

mation is a pair ((G, v), W ) where (G, v ) is a pointed non-

deterministic CGSiand Wis a property of inﬁnite plays

called the winning condition. The strategy problem is, given

such a game, to decide whether there exists a strategy proﬁle

for the grand coalition Ag to enforce the winning condition

against Nature (for more details see, e.g., [4]).

Before stating our meta-theorem we need to introduce a

couple of notions. First we introduce a notion of abstraction

over a group of agents. Informally, abstracting a CGSiG

over an agent consists in erasing her from the group of agents

and letting Nature play for her in G.

Definition 3 (Abstraction). Let A⊆Ag be a group

of agents and let G= (V, E , `, {∼a}a∈Ag)be a CGSi. The

abstraction of Gfrom Ais the nondeterministic CGSiover

set of agents Ag\Adeﬁned as G ↑A:= (V, E 0, `, {∼a}a∈Ag\A),

where for every v∈Vand m∈MAg\A,

(v, m, v0)∈E0if ∃m0∈MAs.t. E(v , (m,m0)) = v0.

Thanks to this notion we can deﬁne the following problem:

Definition 4 (A-strategy problem). The A-strategy

problem takes as input a pointed CGSi(G, v), a set A⊆Ag

of agents and a winning condition W, and returns the an-

swer to the strategy problem for the game ((G ↑Ag\A, v), W ).

The A-strategy problem for (G, v) with winning condition W

thus consists in deciding whether there is a strategy proﬁle

for agents in Ato enforce Wagainst everybody else.

Finally we introduce the following notion, which simply

captures the change of initial position in a game from a

position vto another position v0reachable from v:

Definition 5 (Initial shifting). Let Gbe a CGSiand

let v, v0∈ G . The pointed CGSi(G, v0)is an initial shifting

of (G, v)if v0is reachable from vin G.

We are now ready to state our ﬁrst result.

Theorem 1. If Cis a class of pointed CGSiclosed under

initial shifting and such that the A-strategy problem with ω-

regular objective is decidable on C, then model checking ATL∗

i

is decidable on C.

Proof. Let Cbe such a class of pointed CGSi, and let

(ϕ, (G, v)) be an instance of the model-checking problem for

ATL∗

ion C. A bottom-up algorithm consists in evaluating

each innermost subformula of ϕof the form hAiϕ0, where ϕ0

is thus an LTL formula, on each position v0of Greachable

from v. Evaluating hAiϕ0on v0amounts to solving an in-

stance of the A-strategy problem3with ω-regular objective

(recall that LTL properties are ω-regular). By assumption

(G, v)∈ C, and because Cis closed by initial shifting and

v0is reachable from v, we have that (G, v0)∈ C. Also by

assumption, the A-strategy problem for ω-regular winning

conditions is decidable on C. We thus have an algorithm to

evaluate each hAiϕ0on each v0. One can then mark posi-

tions of the game with fresh atomic propositions indicating

where these formulas hold, and repeat the procedure until

all strategic operators have been eliminated. It then remains

to evaluate a boolean formula in the initial position v.

Let us recall for which classes of nondeterministic CGSi

the strategy problem is known to be decidable. A (nonde-

terministic or deterministic) CGSiGhas hierarchical obser-

vation if there exists a total preorder 4over Ag such that

if a4band v∼av0, then v∼bv0. This notion was reﬁned

in [4] to take into account the agents’ memory, using the

notion of information set: for a ﬁnite play ρ∈Plays(G, v)

and an agent a, the information set of agent aafter ρis

Ia(ρ) := {ρ0∈Plays(G, v)|ρ∼aρ0}. A ﬁnite play ρyields

hierarchical information if there is a total preorder 4over

Ag such that if a4b, then Ia(ρ)⊆Ib(ρ). If all ﬁnite plays

in Plays(G, v) yield hierarchical information for the same

preorder over agents, (G, v) yields static hierarchical infor-

mation. If this preorder can vary depending on the play,

(G, v) yields dynamic hierarchical information. The last gen-

eralisation consists in allowing for transient phases without

hierarchical information: if every inﬁnite play in Plays(G, v)

has inﬁnitely many preﬁxes that yield hierarchical informa-

tion, (G, v) yields recurring hierarchical information.

Proposition 1. Hierarchical observation as well as static,

dynamic and recurring hierarchical information are preserved

by abstraction.

Proof. Abstraction removes agents without aﬀecting ob-

servations of remaining ones. The result thus follows from

the respective deﬁnitions of hierarchical observation and of

static, dynamic and recurring hierarchical information.

Proposition 2. Hierarchical observation as well as static,

dynamic and recurring hierarchical information are preserved

by initial shifting.

This is obvious for hierarchical observation. For the other

cases we establish Lemma 1 below. It is then easy to check

that Proposition 2 holds.

Lemma 1. If a ﬁnite play v·ρ·v0·ρ0yields hierarchical

information in (G, v), so does v0·ρ0in (G, v0), with the same

preorder among agents.

3Observe that if A= Ag then G ↑Ag\A=G, and Nature thus

does not do anything. This is coherent with the fact that

for agents with perfect recall hAgiϕ≡Eϕ, where Eis the

CTL path quantiﬁer, even for imperfect information.

Proof. Assume that v·ρ·v0·ρ0yields hierarchical infor-

mation in (G, v) with preorder 4over Ag. Suppose towards a

contradiction that there are agents a, b ∈Ag such that a4b

but Ia(v0·ρ0)6⊆ Ib(v0·ρ0). This means that there is v0·ρ00 ∈

Plays(G, v0) such that v0·ρ0∼av0·ρ00 but v0·ρ06∼bv0·ρ00.

By deﬁnition of synchronous perfect recall relations we then

have that v·ρ·v0·ρ0∼av·ρ·v0·ρ00 and v·ρ·v0·ρ06∼bv·ρ·v0·ρ00.

This implies that Ia(v·ρ·v0·ρ0)6⊆ Ib(v·ρ·v0·ρ0), which

contradicts the fact that a4b. Therefore for all agents a, b

such that a4bwe have Ia(v0·ρ0)⊆Ib(v0·ρ0), and thus

v0·ρ0yields hierarchical information with preorder 4.

Let Cobs (resp. Cstat,Cdyn ,Crec) be the class of pointed

CGSiwith hierarchical observation (resp. static, dynamic,

recurring hierarchical information). We instantiate Theo-

rem 1 to obtain three decidability results for ATL∗

i.

Theorem 2. Model checking ATL∗

iis decidable on the

class of CGSiwith hierarchical observation.

Proof. By Proposition 2, Cobs is closed under initial shift-

ing. It is proven in [22] that the strategy problem is decid-

able for games with hierarchical observation and ω-regular

objectives. Since, by Proposition 1, all pointed nondeter-

ministic CGSiobtained by abstracting agents from CGSiin

Cobs also yield hierarchical observation, we get that the A-

strategy problem with ω-regular objectives is decidable on

Cobs. We can therefore apply Theorem 1 on Cobs.

It is proven in [4] that the strategy problem with ω-regular

objectives is also decidable for games with static hierarchical

information and for games with dynamic hierarchical infor-

mation. Since Proposition 1 and Proposition 2 also hold for

Cstat and Cdyn, with the same argument as in the proof of

Theorem 2, we obtain the following results as consequences

of Theorem 1:

Theorem 3. Model checking ATL∗

iis decidable on the

class of CGSiwith static hierarchical information.

Theorem 4. Model checking ATL∗

iis decidable on the

class of CGSiwith dynamic hierarchical information.

Note that in fact, since Cobs ⊂ Cstat ⊂ Cdyn, Theorem 2

and Theorem 3 are also obtained as corollaries of Theorem 4,

but we wanted to illustrate how Theorem 1 can be applied

to obtain decidability results for diﬀerent classes of CGSi.

Remark 1. The last result in [4] establishes that the strat-

egy problem is decidable for games with recurring hierarchi-

cal information, but only for observable ω-regular winning

conditions, i.e., when all agents can tell whether a play is

winning or not. Now considering ATL∗

ion Cdyn we could

require atomic propositions to be observable for all agents;

in that case we could evaluate the inner-most strategy quan-

tiﬁers using the above-mentioned result. But then the fresh

atomic propositions that mark positions where these subfor-

mulas hold (see the proof of Theorem 1) would not, in gen-

eral, be observable by all agents. So on Crec we could obtain

a decision procedure for the fragment of ATL∗

iwithout nested

non-trivial strategy quantiﬁers, where “non-trivial” means

for coalitions other than the empty coalition or the one made

of all agents (which, we recall, are simply the CTL path quan-

tiﬁers). We do not state it explicitly due to lack of space and

because it does not seem of much interest.

Concerning complexity, the strategy problem for games

with imperfect information and hierarchical observation is

already nonelementary [29, 27], hence the following result:

Corollary 1. Model checking ATL∗

iis nonelementary

on games with hierarchical observation, hence also for games

with static or dynamic hierarchical information.

We now turn to ATL with imperfect information and strat-

egy context, and study its model-checking problem.

4. ATLiWITH STRATEGY CONTEXT

While in ATL strategies for all agents are forgotten each

time a new strategy quantiﬁer is met, in ATL with strat-

egy context (ATLsc ) [5, 9, 24] agents keep using the same

strategy as long as the formula does not say otherwise. In

this section we consider ATLsc with imperfect information

(ATLsc,i). As far as we know, the only existing work on

this logic is [25], which proved its model-checking problem

to be decidable in the case where all agents have the same

observation of the game. We extend signiﬁcantly this result

by establishing that the model-checking problem is decid-

able as long as strategy quantiﬁcation is hierarchical, in the

sense that if there is a strategy quantiﬁcation for agent a

nested in a strategy quantiﬁcation for agent b, then bshould

observe no more than a. In other terms, innermost strategic

quantiﬁcations should concern agents who observe more.

4.1 Syntax and semantics

The models are still CGSi. To remember which agents are

currently bound to a strategy, and what these strategies are,

the semantics uses strategy contexts. Formally, a strategy

context for a set of agents B⊆Ag is a strategy proﬁle σB.

We deﬁne the composition of strategy contexts as follows.

If σBis a strategy context for Band σAis a new strategy

proﬁle for coalition A, we let σA◦σBbe the strategy context

for A∪Bdeﬁned as σA∪B:a7→ (σA(a) if a∈A,

σB(a) otherwise .

So if ais assigned a strategy by σA, her strategy in σA◦σB

is σA(a). If she is not assigned a strategy by σAher strategy

remains the one given by σB, if any.

Also, given a strategy context σBand a set of agents

A⊆Ag, we let (σB)\Abe the strategy context obtained

by restricting σBto the domain B\A.

Finally, because agents who do not change their strategy

keep playing the one they were assigned, if any, we cannot

forget the past at each strategy quantiﬁer, as in the seman-

tics of ATL∗

i(see Section 3.1). We thus deﬁne the outcome of

a strategy proﬁle σAafter a ﬁnite play ρ, written Out(ρ, σA),

as the set of inﬁnite plays πthat start with ρand then fol-

low σA:π∈Out(ρ, σA) if π=ρ·π0for some π0, and for all

i≥ |ρ| − 1, there exists a joint move m∈MAg such that

E(πi,m) = πi+1 and for each a∈A,ma=σa(π[0, i]).

To diﬀerentiate from ATL∗, in ATL∗

sc the strategy quanti-

ﬁer for a coalition Ais written h·A·i instead of hAi.ATL∗

sc

also has an additional operator, (|A|), that releases agents in

Afrom their current strategy, if they have one. The syntax

of ATL∗

sc,i is the same as that of ATL∗

sc and is thus given by

the following grammar:

ϕ::= p| ¬ϕ|ϕ∨ϕ| h·A·iϕ|(|A|)ϕ|Xϕ|ϕUϕ,

where p∈ AP and A⊆Ag. We use standard abbreviations:

>:= p∨ ¬p,⊥:= ¬>,Fϕ:= >Uϕ, and Gϕ:= ¬F¬ϕ.

Remark 2. In [24] the syntax of ATL∗

sc contains in ad-

dition operators h·A·i and (|A|)for complement coalitions.

While they add expressivity when the set of agents is not

ﬁxed, and are thus of interest when considering expressiv-

ity or satisﬁability, they are redundant if we consider model

checking, which is our case in this work. To simplify pre-

sentation we thus choose not to consider them here.

The semantics of ATL∗

sc,i is deﬁned with regards to a CGSi

G= (V, E , `, {∼a}a∈Ag), an inﬁnite play π, a position i∈N

along this play, and a strategy context σB. The semantics

is deﬁned by induction on formulas:

G, π, i |=σBpif p∈`(πi)

G, π, i |=σB¬ϕif G, π, i 6|=σBϕ

G, π, i |=σBϕ∨ϕ0if G, π, i |=σBϕor G, π, i |=σBϕ0

G, π, i |=σBh·A·iϕif there exists a strategy proﬁle σAs.t.

for all π0∈Out(π[0, i], σA◦σB),

G, π0, i |=σA◦σBϕ

G, π, i |=σB(|A|)ϕif G, π, i |=(σB)\Aϕ

G, π, i |=σBXϕif G, π, i + 1 |=σBϕ

G, π, i |=σBϕUϕ0if there exists j≥is.t. G, π, j |=σBϕ0

and, for all ksuch that i≤k < j,

G, π, k |=σBϕ.

The notion of closed formula is as deﬁned in Section 3.1

and once more, the semantics of a closed formula ϕbeing

independent from the future, we may write G, v |=σBϕin-

stead of G, π, 0|=σBϕfor any inﬁnite play πthat starts in

position v. We also write G, v |=ϕif G, v |=σ∅ϕ, that is if

ϕholds in vwith the empty strategy context.

The model-checking problem for ATL∗

sc,i consists in decid-

ing, given a closed ATL∗

sc,i formula ϕand a ﬁnite pointed

CGSi(G, v), whether G, v |=ϕ.

We now present QCTL∗with imperfect information, or

QCTL∗

ifor short, before proving our main result on the

model-checking problem for ATL∗

sc,i by reducing it to the

model-checking problem for a decidable fragment of QCTL∗

i.

4.2 QCTL∗with imperfect information

Quantiﬁed CTL∗, or QCTL∗for short, is an extension of

CTL∗with second-order quantiﬁers on atomic propositions

that has been well studied [33, 20, 21, 23]. It has recently

been further extended to take into account imperfect infor-

mation, resulting in the logic called QCTL∗with imperfect

information, or QCTL∗

i[3]. We brieﬂy present this logic, as

well as a decidability result on its model-checking problem

proved in [3] and that we rely on to establish our result on

the model checking of ATL∗

sc,i.

Imperfect information is incorporated into QCTL∗by con-

sidering Kripke models with internal structure in the form of

local states, like in distributed systems (see for instance [17]),

and then parameterising quantiﬁers on atomic propositions

with observations that deﬁne what portions of the states

a quantiﬁer can “observe”. The semantics is then adapted

to capture the idea of quantiﬁcation on atomic propositions

being made with partial observation.

Let us ﬁx a collection {Li}i∈[n]of ndisjoint ﬁnite sets of

local states. We also let Xn=L1×. . . ×Ln.

Definition 6. Acompound Kripke structure (CKS) over

AP is a Kripke structure S= (S, R, `)such that S⊆Xn.

The syntax of QCTL∗

iis that of QCTL∗, except that quan-

tiﬁers over atomic propositions are parameterised by a set

of indices that deﬁnes what local states the quantiﬁer can

“observe”. It is thus deﬁned by the following grammar:

ϕ:= p| ¬ϕ|ϕ∨ϕ|Eϕ| ∃op. ϕ |Xϕ|ϕUϕ

where p∈ AP and o⊂Nis a ﬁnite set of indices. As usual,

we let Aϕ:= ¬E¬ϕ.

A ﬁnite set o⊂Nis called an observation, and two states

s= (l1,...,ln) and s0= (l0

1,...,l0

n) are o-indistinguishable,

written s≈os0, if for all i∈[n]∩o, it holds that li=l0

i.

The intuition is that a quantiﬁer with observation omust

choose the valuation of atomic propositions uniformly with

respect to o. Note that in [3], two semantics are considered

for QCTL∗

i, just like in [23] for QCTL∗: the structure se-

mantics and the tree semantics. In the former, formulas are

evaluated directly on the structure, while in the latter the

structure is ﬁrst unfolded into an inﬁnite tree. Here we only

present the tree semantics, as it is this one that allows us to

capture agents with perfect recall. But we ﬁrst need a few

more deﬁnitions.

For p∈ AP, two labelled trees t= (τ, `) and t0= (τ0, `0)

are equivalent modulo p, written t≡pt0, if τ=τ0and for

each node u∈τ,`(u)\ {p}=`0(u)\ {p}. So t≡pt0if they

are the same trees, except for the labelling of proposition p.

This notion of equivalence modulo pis the one used to

deﬁne quantiﬁcation on atomic propositions in QCTL∗: in-

tuitively, an existential quantiﬁcation over pchooses a new

labelling for valuation p, all else remaining the same, and the

evaluation of the formula continues from the current node

with the new labelling. For imperfect information we need

to express the fact that this new labelling for a proposition is

done uniformly with regards to the quantiﬁer’s observation.

First, we deﬁne the notion of indistinguishability between

two nodes in the unfolding of a CKS. Let obe an observation,

let τbe an Xn-tree (which may be obtained by unfolding

some pointed CKS), and let u=s0. . . siand u0=s0

0. . . s0

jbe

two nodes in τ. The nodes uand u0are o-indistinguishable,

written u≈ou0, if i=jand for all k∈ {0,...,i}, we have

sk≈os0

k. Observe that this deﬁnition corresponds to the no-

tion of synchronous perfect recall in CGSi(see Section 3.1).

We now deﬁne what it means for the labelling of an atomic

proposition to be uniform with regards to an observation.

Definition 7. Let t= (τ, `)be a labelled Xn-tree, let

p∈ AP be an atomic proposition and o⊂Nan observation.

Tree tis o-uniform in pif for every pair of nodes u, u0∈τ

such that u≈ou0, we have p∈`(u)iﬀ p∈`(u0).

The satisfaction relation |=t(tis for tree semantics) is now

deﬁned as follows, where t= (τ, `) is a labelled Xn-tree, λ

is a path in τand i∈Na position along that branch:

t, λ, i |=tpif p∈`(λi)

t, λ, i |=t¬ϕif t, λ, i 6|=tϕ

t, λ, i |=tϕ∨ϕ0if t, λ, i |=tϕor t, λ, i |=tϕ0

t, λ, i |=tEϕif there exists λ0∈P aths(λi)

such that t, λ0,0|=tϕ

t, λ, i |=t∃op. ϕ if there exists t0≡ptsuch that

t0is o-uniform in pand t0, λ, i |=tϕ

t, λ, i |=tXϕif t, λ, i + 1 |=tϕ

t, λ, i |=tϕUϕ0if there exists j≥isuch that t, λ, j |=tϕ0

and for i≤k < j, t, λ, j |=tϕ

Similarly to ATL∗

iand ATL∗

sc,i, we say that a QCTL∗

ifor-

mula is closed if all temporal operators are in the scope of a

path quantiﬁer. The semantics of such formulas depending

only on the current node, for a closed formula ϕwe may

write t|=tϕfor t, r |=tϕ, where ris the root of t, and

given a CGSiG, a state sand a QCTL∗

iformula ϕ, we write

S, s |=tϕif tS(s)|=tϕ.

Remark 3. In [3] the syntax is presented with path for-

mulas distinguished from state formulas, and the semantics

is deﬁned accordingly. To make the presentation more uni-

form with that of ATLsc,i we chose here a diﬀerent, but equiv-

alent, presentation.

Remark 4. Note that when nis ﬁxed, the propositional

quantiﬁer with perfect information from QCTL∗is equivalent

to the QCTL∗

iquantiﬁer that observes all the components,

i.e., the quantiﬁer parameterised with observation [n].

The model-checking problem for QCTL∗

iis the following:

given a closed QCTL∗

iformula ϕand a ﬁnite pointed CKS

(S, s), decide whether S, s |=tϕ.

We now deﬁne the class of QCTL∗

iformulas for which the

model-checking problem is known to be decidable with the

tree semantics.

Definition 8. AQCTL∗

iformula ϕis hierarchical if for

all subformulas ϕ1, ϕ2of the form ϕ1=∃o1p1. ϕ0

1and ϕ2=

∃o2p2. ϕ0

2where ϕ2is a subformula of ϕ0

1, we have o1⊆o2.

The following result is proved in [3], where QCTL∗

i,⊂is the

set of hierarchical QCTL∗

iformulas:

Theorem 5 ([3]). Model checking QCTL∗

i,⊂with tree

semantics is decidable.

4.3 Model checking ATL∗

sc,i

We establish that model checking ATL∗

sc,i is decidable on

a class of instances whose deﬁnition relies on the notion of

hierarchical observation.

Definition 9. Let G= (V, E, `, {∼a}a∈Ag )be a CGSi,

and let a, b ∈Ag be two agents. Agent aobserves no more

than agent bin G, written a4Gb, if for every pair of po-

sitions v, v0∈V,v∼bv0implies v∼av0. We say that

A⊆Ag is hierarchical in Gif 4Gis a total preorder on A.

If a set of agents Ais hierarchical in a CGSiG, we thus

may talk about maximal and minimal agents in A, referring

to maximal and minimal elements of Afor the relation 4G.

The essence of the requirement that makes the problem

decidable is the same as for the decidability result on QCTL∗

i

(Theorem 5): nesting of quantiﬁers (here, strategy quan-

tiﬁers) should be hierarchical, with those observing more

inside those observing less. However, unlike in QCTL∗

i, in

ATL∗

sc,i observations are not part of formulas, but rather

they are given by the models. We thus deﬁne the notion of

hierarchical ATL∗

sc,i formula with respect to a given CGSi:

Definition 10. Let Φbe an ATL∗

sc,i formula and Ga

CGSi. We say that Φis hierarchical in Gif:

•for every subformula ϕof the form ϕ=h·A·iϕ0,Ais

hierarchical in G, and

•for all subformulas ϕ1, ϕ2of the form ϕ1=h·A1·iϕ0

1

and ϕ2=h·A2·iϕ0

2where ϕ2is a subformula of ϕ0

1,

maximal agents of A1observe no more than minimal

agents of A2.

An instance (Φ,(G, v)) of the model-checking problem for

ATL∗

sc,i is hierarchical if Φis hierarchical in G.

In the rest of the section we establish the following:

Theorem 6. Model checking ATL∗

sc,i is decidable on the

class of hierarchical instances.

We build upon the proof in [24] that establishes the decid-

ability of the model-checking problem for ATL∗

sc by reduc-

tion to the model-checking problem for QCTL∗. The main

diﬀerence is that we reduce to the model-checking problem

for QCTL∗

iinstead, using quantiﬁers parameterised with ob-

servations corresponding to agents’ observations. We also

need to make a couple of adjustments to obtain formulas in

the decidable fragment QCTL∗

i,⊂.

Let (Φ,(G, vι)) be a hierarchical instance of the ATL∗

sc,i

model-checking problem, where G= (V, E, `, {∼a}a∈Ag ) is a

CGSiover AP. In the reduction we will transform Φ into

an equivalent QCTL∗

iformula Φ0in which we need to refer

to the current position in the model G, and also to talk

about moves taken by agents. To do so, we consider the

additional sets of atomic propositions APv:= {pv|v∈V}

and APm:= {pa

m|a∈Ag and m∈M}, that we take

disjoint from AP.

First we deﬁne the CKS SGon which Φ0will be evalu-

ated. Since the models of the two logics use diﬀerent ways

to represent imperfect information (equivalence relations on

positions for CGSiand local states for CKS) this requires a

bit of work. First, for each v∈Vand a∈Ag, let us de-

ﬁne [v]aas the equivalence class of vfor relation ∼a. Now,

noting Ag = {a1,...,an}, we deﬁne for each i∈[n] the set

Li:= {[v]ai|v∈V}of local states for agent ai. Since we

need to know the actual position of the CGSito deﬁne the

dynamics, we also let Ln+1 := V. States of SGwill thus

be tuples in L1×. . . ×Ln×Ln+1. For each v∈ G, let

sv:= ([v]a1,...,[v]an, v) be its corresponding state in SG.

We can now deﬁne SG:= (S, R, `0), where

•S:= {sv|v∈V},

•R:= {(sv, sv0)| ∃m∈MAg s.t. E(v, m) = v0}, and

•`0(sv) := `(v)∪ {pv}.

To make the connection between ﬁnite plays in Gand

nodes in tree unfoldings of SG, let us deﬁne, for every ﬁnite

play ρ=v0. . . vk, the node uρ:= sv0. . . svkin tSG(sv0)

(which exists, by deﬁnition of SGand of tree unfoldings).

Observe that the mapping ρ7→ uρis in fact a bijection

between the set of ﬁnite plays starting in a given position v

and the set of nodes in tSG(sv).

Now it should be clear that giving to a propositional quan-

tiﬁer in QCTL∗

iobservation oi:= {i}, for i∈[n], amounts to

giving him the same observation as agent ai. Formally, one

can prove the following lemma, simply by applying the def-

initions of observational equivalence in the two frameworks:

Lemma 2. For all ﬁnite plays ρ, ρ0starting in position v,

ρ∼aiρ0iﬀ uρ≈oiuρ0in tSG(sv).

We now describe the translation4from ATLsc,i formulas

to QCTL∗

iformulas. First we recall the translation from [24]

for the perfect-information case.

The translation from ATLsc to QCTL∗is parameterised

by a coalition B⊂Ag, that conveys the set of agents who

4Here we abuse language: the construction depends on the

model Gand is therefore not a translation in the usual sense.

are currently bound to a strategy. It is deﬁned by induction

on Φ as follows:

pB:= p¬ϕB:= ¬ϕB

ϕ∨ϕ0B:= ϕB∨ϕ0B(|A|)ϕB:= ϕB\A

XϕB:= XϕBϕUϕ0B:= ϕBUϕ0B

The only non-trivial case is for formulas of the form h·A·iϕ.

For the rest of the section, we let M = {m1,...,ml}. Now,

if A={ai1,...,aik}, we deﬁne

h·A·iϕB:= ∃mai1

1. . . mai1

l. . . maik

1. . . maik

lpout.

Φstrat(A)∧Φout (A∪B)∧A(Gpout →ϕA∪B),

where

Φstrat(A) := ^

a∈A

AG _

m∈M

(ma∧^

m06=m

¬m0a)

and

Φout(A) := pout ∧AG [¬pout →AX¬pout]∧AG

pout →

_

v∈V_

m∈MA

pv∧pm∧AX

_

v0∈E(v,m)

pv0↔pout

.

In Φout(A), for m= (ma)a∈A∈MA, notation pmstands

for the propositional formula Va∈Ama

awhich characterizes

the joint move mthat agents in Aplay in v. Also, E(v , m)

is the set of possible next positions when the current one is

vand agents in Aplay m, and it is deﬁned as E(v, m) :=

{E(v, (m,m0)) |m0∈MAg\A}.

The idea of this translation is the following: ﬁrst, for each

agent a∈Aand each possible move m∈M, an existential

quantiﬁcation on the atomic proposition ma“chooses” for

each ﬁnite play ρof (G, vι) (or, equivalently, for each node

uρof tSG(svι)) whether agent aplays move min ρor not,

coded by mabeing chosen to be true a in ρor not. Formula

Φstrat(A) ensures that each agent achooses exactly one move

in each ﬁnite play, and thus that atomic propositions ma

characterise a strategy for her. An atomic proposition pout

is then used to mark the paths that follow the currently

ﬁxed strategies: formula Φout(A∪B) states that pout marks

exactly the outcome of strategies just chosen for agents in A,

as well as those of agents in B, that were chosen previously

by a strategy quantiﬁer “higher” in Φ.

Note that we simpliﬁed slightly Φstrat(A) and Φout(A),

using the fact that unlike in [24], we have assumed in our

deﬁnition of CGSithat the set of available moves is the same

for all agents in all positions (see Footnote 2).

It is proven in [24] that this translation is correct, in the

sense that for every ATLsc closed formula ϕand pointed

perfect-information concurrent game structure (G, v), letting

SGbe as described above but removing the local states for

all agents and keeping only the Ln+1 component, we have:

G, v |=ϕiﬀ tSG(sv)|=tϕ∅.

We now explain how to adapt this translation to the case of

imperfect information. Observe that the only diﬀerence be-

tween ATL∗

sc and ATL∗

sc,i is that in the latter, strategies must

be deﬁned uniformly over indistinguishable ﬁnite plays, i.e.,

a strategy σfor an agent amust be such that if ρ∼aρ0, then

σ(ρ) = σ(ρ0). To enforce that the strategies coded by atomic

propositions main h·A·iϕBare uniform, we use the propo-

sitional quantiﬁers with partial observation of QCTL∗

i. For-

mally, we deﬁne a translation fBfrom ATL∗

sc,i to QCTL∗

i.

It is deﬁned exactly as the one from ATL∗

sc to QCTL∗, except

for the following inductive case.

If A={ai1,...,aik}we let

^

h·A·iϕ

B

:= ∃oi1mai1

1. . . mai1

l. . . ∃oikmaik

1. . . maik

l∃pout.

Φstrat(A)∧Φout (A∪B)∧A(Gpout →eϕA∪B),

where Φstrat(A) and Φout (A) are deﬁned as before, and ∃pout

is a macro for ∃{1,...,n+1}pout (see Remark 4).

So the only diﬀerence from the previous translation is

that now, the labelling of each atomic proposition maimust

be oi-uniform. This means that if two nodes uand u0in

tSG(svι) are oi-indistinguishable, then uis labelled with mai

if and only if u0also is. In other words, in the strategy coded

by atomic propositions mai, agent aiplays min uif and only

if she also plays it in u0, and thus this strategy is uniform

(recall that, by Lemma 2, observation oicorrectly reﬂects

agent ai’s observation in tSG(svι)). It is then clear that this

translation is correct:

G, vι|= Φ iﬀ tSG(svι)|=te

Φ∅.(1)

However, even if we have taken (Φ,(G, vι)) to be a hierar-

chical instance, e

Φ∅is not in the decidable fragment QCTL∗

i,⊂.

Indeed, with the current deﬁnition of observations {oi}i∈[n],

hierarchical observation in Gdoes not imply hierarchical ob-

servation in SG: since oi={i}, for i6=jit is never the

case that oi⊆oj. Still, we note that if agent ajobserves no

more than agent ai, then letting aisee also what agent aj

sees does not increase her knowledge of the situation:

Lemma 3. If aj4Gai, then for all ﬁnite plays ρ, ρ0that

start in the same position, uρ≈oiuρ0iﬀ uρ≈oi∪ojuρ0.

Proof. Assume that aj4Gai. It is enough to see that

for every pair of states sv, sv0in SG, we have sv≈oisv0iﬀ

sv≈oi∪ojsv0. The right-to-left implication is obvious: if two

states have the same i-th and j-th components, in particular

they have the same i-th component. For the other direction,

assume that sv≈oisv0. This means that [v]ai= [v0]ai,

and thus that v∼aiv0. Since aj4Gai, we also have that

v∼ajv0, and thus that [v]aj= [v0]aj, and it follows that

sv≈oi∪ojsv0.

In the light of this Lemma 3, we can safely redeﬁne obser-

vations as follows: for each i∈[n], we let

o0

i:= [

j|aj4Gai

oj.

Observe that in fact o0

i={j|aj4Gai}. Informally, a

quantiﬁer with observation o0

isees what agent aiobserves

(note that 4Gis reﬂexive), as well as what agents that see

no more than aiobserve.

Let us deﬁne a new version of the translation fB. First,

Φ being hierarchical in G, for each subformula of Φ of the

form h·A·iϕwe have that Ais hierarchical in G. It is thus pos-

sible to choose for agents in Aan indexing A={ai1,...,aik}

such that for all 1 ≤c < d ≤k, we have aic4Gaid.

Now the translation remains the same as before except for

the following inductive case:

If A={ai1,...,aik}, where for all 1 ≤c < d ≤k, we

have aic4Gaid, we let

^

h·A·iϕ

B

:= ∃o0

i1mai1

1. . . mai1

l. . . ∃o0

ikmaik

1. . . maik

l∃pout.

Φstrat(A)∧Φout (A∪B)∧A(Gpout →eϕA∪B),

where Φstrat(A) and Φout (A) are deﬁned as before.

From Lemma 3 we have that this new translation is still

correct in the sense of Equation (1). In addition, for all

1≤c < d ≤kwe have o0

ic⊆o0

id.

Now consider formula e

Φ∅. Because Φ is hierarchical in G,

for every pair of subformulas ϕ1, ϕ2of the form ϕ1=h·A1·iϕ0

1

and ϕ2=h·A2·iϕ0

2where ϕ2is a subformula of ϕ0

1, maximal

agents of A1observe no more than minimal agents of A2.

It is then easy to see that e

Φ∅would be hierarchical if there

were not the perfect-information quantiﬁcations on atomic

proposition pout that break the monotony of observations

along subformulas when there are nested strategic quanti-

ﬁers. We explain how to remedy this last problem.

We remove altogether proposition pout, and we use instead

the formula ψout(A) deﬁned below to characterise which

paths are in the outcome of the currently-ﬁxed strategies:

ψout(A) := G

^

v∈V^

m∈MA

pv∧pm→X_

v0∈E(v,m)

pv0

.

Clearly, this formula holds in a path λof tSG(svι) marked

with propositions macharacterising strategies for agents in

A, if at each point along λcorresponding to some position

v, the next point in λcorresponds to a position v0that can

be attained from vwhen agents in Aeach play the move

prescribed by their current strategy. The last modiﬁcation

to fBis thus the following:

If A={ai1,...,aik}, where for all 1 ≤c < d ≤k, we

have aic4Gaid, we let

^

h·A·iϕ

B

:= ∃o0

i1mai1

1. . . mai1

l. . . ∃o0

ikmaik

1. . . maik

l.

Φstrat(A)∧Aψout (A∪B)→eϕA∪B,

where Φstrat(A) is deﬁned as before.

It follows from the above considerations that this transla-

tion is still correct in the sense of Equation (1), and one can

check that e

Φ∅is a hierarchical QCTL∗

iformula. We conclude

the proof by recalling that by Theorem 5, model checking

QCTL∗

i,⊂is decidable.

Concerning complexity, model checking ATLsc being al-

ready nonelementary [24], so is it for ATLsc,i .

5. CONCLUSION

In this work we established new decidability results for

the model-checking problem of ATL∗with imperfect infor-

mation and perfect recall as well as its extension with strat-

egy context. Should new decidable classes of multiplayer

games with imperfect information be discovered, and assum-

ing the reasonable property of closure under initial shifting,

our transfer theorem (Theorem 1) would entail new decid-

ability results also for ATL∗

i. As for ATL∗

sc,i, it would be

interesting to investigate whether a meaningful notion of

hierarchical instances based on, e.g., dynamic or recurring

hierarchical information instead of hierarchical observation

as here, could lead to stronger decidability results.

REFERENCES

[1] T. Agotnes, V. Goranko, and W. Jamroga.

Alternating-Time Temporal Logics with Irrevocable

Strategies. In TARK, pages 15–24, 2007.

[2] R. Alur, T. Henzinger, and O. Kupferman.

Alternating-Time Temporal Logic. J. ACM,

49(5):672–713, 2002.

[3] R. Berthon, B. Maubert, and A. Murano. Quantiﬁed

CTL with imperfect information. CoRR,

abs/1611.03524, 2016.

[4] D. Berwanger, A. B. Mathew, and M. van den

Bogaard. Hierarchical information patterns and

distributed strategy synthesis. In Automated

Technology for Veriﬁcation and Analysis - 13th

International Symposium, ATVA 2015, Shanghai,

China, October 12-15, 2015, Proceedings, pages

378–393, 2015.

[5] T. Brihaye, A. D. C. Lopes, F. Laroussinie, and

N. Markey. ATL with strategy contexts and bounded

memory. In Logical Foundations of Computer Science,

International Symposium, LFCS 2009, Deerﬁeld

Beach, FL, USA, January 3-6, 2009. Proceedings,

pages 92–106, 2009.

[6] N. Bulling and W. Jamroga. Comparing variants of

strategic ability: how uncertainty and memory

inﬂuence general properties of games. Autonomous

Agents and Multi-Agent Systems, 28(3):474–518, 2014.

[7] E. Clarke and E. Emerson. Design and Synthesis of

Synchronization Skeletons Using Branching-Time

Temporal Logic. In 81, LNCS 131, pages 52–71, 1981.

[8] E. Clarke, O. Grumberg, and D. Peled. Model

Checking. 2002.

[9] A. Da Costa, F. Laroussinie, and N. Markey. ATL

with Strategy Contexts: Expressiveness and Model

Checking. In FSTTCS’10, LIPIcs 8, pages 120–132,

2010.

[10] C. Dima, C. Enea, and D. P. Guelev. Model-checking

an alternating-time temporal logic with knowledge,

imperfect information, perfect recall and

communicating coalitions. In Proceedings First

Symposium on Games, Automata, Logic, and Formal

Veriﬁcation, GANDALF 2010, Minori (Amalﬁ Coast),

Italy, 17-18th June 2010., pages 103–117, 2010.

[11] C. Dima and F. L. Tiplea. Model-checking ATL under

imperfect information and perfect recall semantics is

undecidable. CoRR, abs/1102.4225, 2011.

[12] R. Fagin, J. Y. Halpern, Y. Moses, and M. Y. Vardi.

Reasoning about knowledge, volume 4. MIT press

Cambridge, 1995.

[13] B. Finkbeiner and S. Schewe. Uniform distributed

synthesis. In 20th IEEE Symposium on Logic in

Computer Science (LICS 2005), 26-29 June 2005,

Chicago, IL, USA, Proceedings, pages 321–330, 2005.

[14] T. French. Decidability of quantifed propositional

branching time logics. In Australian Joint Conference

on Artiﬁcial Intelligence, pages 165–176. Springer,

2001.

[15] D. P. Guelev and C. Dima. Model-checking strategic

ability and knowledge of the past of communicating

coalitions. In Declarative Agent Languages and

Technologies VI, 6th International Workshop, DALT

2008, Estoril, Portugal, May 12, 2008, Revised

Selected and Invited Papers, pages 75–90, 2008.

[16] D. P. Guelev, C. Dima, and C. Enea. An

alternating-time temporal logic with knowledge,

perfect recall and past: axiomatisation and

model-checking. Journal of Applied Non-Classical

Logics, 21(1):93–131, 2011.

[17] J. Y. Halpern and M. Y. Vardi. The complexity of

reasoning about knowledge and time. i. lower bounds.

Journal of Computer and System Sciences,

38(1):195–237, 1989.

[18] W. Jamroga and W. van der Hoek. Agents that Know

How to Play. 63(2-3):185–219, 2004.

[19] P. Kazmierczak, T. ˚

Agotnes, and W. Jamroga.

Multi-agency is coordination and (limited)

communication. In PRIMA 2014: Principles and

Practice of Multi-Agent Systems - 17th International

Conference, Gold Coast, QLD, Australia, December

1-5, 2014. Proceedings, pages 91–106, 2014.

[20] O. Kupferman. Augmenting branching temporal logics

with existential quantiﬁcation over atomic

propositions. In CAV’95, LNCS 939, pages 325–338.

Springer, 1995.

[21] O. Kupferman, P. Madhusudan, P. S. Thiagarajan,

and M. Y. Vardi. Open systems in reactive

environments: Control and synthesis. In CONCUR’00,

LNCS 1877, pages 92–107. Springer, 2000.

[22] O. Kupferman and M. Y. Vardi. Synthesizing

distributed systems. In 16th Annual IEEE Symposium

on Logic in Computer Science, Boston, Massachusetts,

USA, June 16-19, 2001, Proceedings, pages 389–398,

2001.

[23] F. Laroussinie and N. Markey. Quantiﬁed CTL:

expressiveness and complexity. Logical Methods in

Computer Science, 10(4), 2014.

[24] F. Laroussinie and N. Markey. Augmenting ATL with

strategy contexts. Inf. Comput., 245:98–123, 2015.

[25] F. Laroussinie, N. Markey, and A. Sangnier. ATLsc

with partial observation. In Proceedings Sixth

International Symposium on Games, Automata, Logics

and Formal Veriﬁcation, GandALF 2015, Genoa,

Italy, 21-22nd September 2015., pages 43–57, 2015.

[26] M. Pauly. A Modal Logic for Coalitional Power in

Games. 12(1):149–166, 2002.

[27] G. Peterson, J. Reif, and S. Azhar. Lower bounds for

multiplayer noncooperative games of incomplete

information. Computers & Mathematics with

Applications, 41(7):957–992, 2001.

[28] G. Peterson, J. Reif, and S. Azhar. Decision

algorithms for multiplayer noncooperative games of

incomplete information. Computers & Mathematics

with Applications, 43(1):179–206, 2002.

[29] A. Pnueli and R. Rosner. Distributed reactive systems

are hard to synthesize. In 31st Annual Symposium on

Foundations of Computer Science, St. Louis,

Missouri, USA, October 22-24, 1990, Volume II, pages

746–757, 1990.

[30] J. Queille and J. Sifakis. Speciﬁcation and Veriﬁcation

of Concurrent Programs in Cesar. In 81, LNCS 137,

pages 337–351, 1981.

[31] S. Schewe and B. Finkbeiner. Distributed synthesis for

alternating-time logics. In Automated Technology for

Veriﬁcation and Analysis, 5th International

Symposium, ATVA 2007, Tokyo, Japan, October

22-25, 2007, Proceedings, pages 268–283, 2007.

[32] P. Schobbens. Alternating-Time Logic with Imperfect

Recall. ENTCS, 85(2):82–93, 2004.

[33] A. Sistla. Theoretical Issues in the Design and

Ceriﬁcation of Distributed Systems. PhD thesis,

Harvard University, Cambridge, MA, USA, 1983.

[34] D. Walther, W. van der Hoek, and M. Wooldridge.

Alternating-Time Temporal Logic with Explicit

Strategies. In TARK, pages 269–278, 2007.